INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2003; 00:1–20 Prepared using fldauth.cls [Version: 2002/09/18 v1.01] Adaptive Discontinuous Galerkin Method for the Shallow Water Equations Jean-Fran¸ cois Remacle 1, ∗ , Sandra Soares Fraz˜ ao 1, , 3 Xiangrong Li 2 and Mark S. Shephard 2 1 Department of Civil Engineering, Place du Levant 1, 1348 Louvain-la-Neuve, Belgium 2 Scientific Computation Research Center, CII-7011, 110 8th Street, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, U.S.A. 3 Fonds National de la Recherche Scientifique,rue dEgmont 5 B - 1000 Bruxelles. SUMMARY In this paper, we present a Discontinuous Galerkin formulation of the shallow water equations. An orthogonal basis is used for the spatial discretization and an explicit Runge-Kutta scheme is used for time discretization. Some results of second order anisotropic adaptive calculations are presented for dam breaking problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities like hydraulic jumps. Copyright c  2003 John Wiley & Sons, Ltd. key words: Shallow Water Equations, Anisotropic Meshes, Discontinuous Galerkin Method 1. Introduction The Discontinuous Galerkin Method (DGM) was initially introduced by Reed and Hill in 1973 [19] as a technique to solve neutron transport problems. Recently, the DGM has become popular and it has been used for solving a wide range of problems [5]. The DGM is a finite element method in the sense that it involves a double discretization. First, the physical domain Ω is discretized into a collection of N e elements T e = Ne  i=1 Ω i (1) called a mesh. Then, the continuous function space V (Ω) containing the solution u of a given PDE is approximated using a finite expansion into polynomials in space and finite differences in time. The accuracy of a finite element discretization depends both on geometrical and functional discretizations. Adaptivity seeks an optimal combination of these two ingredients: ∗ Correspondence to: remacle@gce.ucl.ac.be Copyright c  2003 John Wiley & Sons, Ltd.